Q&A: How do I solve a system of equations?

You need independent equations to obtain a specific answer


How can I solve the following system of equations?

$$ 3x - 2y = 6 $$ $$ 6x - 4y = 12 $$


A system of equations is a set of two separate equations. We usually want them to be independent so that we can solve for both variables. Independent means (in this case) that they are not multiples of each other and are in fact two separate equations. On a graph they would be two different lines that intersect at a single point.

Why two equations? Well, look at either equation and try to solve for both variables. The best you can do is to write one in terms of the other, but you will never be able to determine what each variable is unless you have a second equation. Because there are two unknowns, we need two equations.

However, any method of solving systems of equations will fail in this case. Why? These are not independent equations. In fact, if you look carefully you will find that the bottom equation is just 2 times the top equation. That is a dead giveaway that you won't be able to solve for a unique (x,y) answer.

In short: The answer to your problem is that you can't solve that set of linear equations. Since they both represent the same line you will have infinitely many places that the two lines intersect.